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???1?,?v?36m/s
Thus, we can verify that the accuracy of our model will not matter with the simplifi-cation of ocean currents.
6 Modeling For Optimization of Search Plan
We need to know key parameters of kinds of planes to determine the most optimized scheduling solution of the searching zone. Whatever the shape of searching zone, we all use the probability calculated by the Bayesian model to act as weight coefficient. Then we multiply the coefficient with the corresponding area to get the equivalent area. The scheduling scheme should meet two condition: (1) Searching planes must cover the searching area. (2) We should select proper kind and quantity of searching plane. Thus, we obtain the key factors to solution – function expression and parame-ters. By using the Dinkelbach and NonConvexDivide algorithm we can work out the most optimized solution with high efficiency.
6.1 The Global Optimization Model
6.1.1 Preparation of the Model
? Considering maximum speed, search ability, maximum battery life and initial
distance from searching area of professional search and rescue plane differs from each other, we establish a mathematical model as follows:
The time of plane i to make a round-trip at full speed is
?a?2Da/V?aTiii
The time of plane i to implement operations in the sea to be searching zone is
?aL?aTi?Ti?Ti The whole mobilized number of plane i is
Li?T/Ti
The equivalent area of the searching zone can be expressed as
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sX?arctan?SsY?S?4sX?0?S0??S0?sX?sY??180?S?p?1??ST???S?p?(ST?S)?(1?p)ST??? ?
? The goals of the model is to confirm the best of search and rescue plane to make a complete search for the entire zone with the least amount of time T.
Given this ,we introduce the following decision variables:
1, plane ai take part in the searching action 0, plane ai will not take part in the action
To implement search coverage rapidly and efficiently in the searching zone , we need to analyze air forces in the composition of the time used in the search operation.
We assume the whole operation begins at ts, ends at te, so the time of the
whole action is T = te - ts. Only planes that arrive at searching zone before the end of the operation can participate in action. Every plane ai has the opportu-nity to take part in the search. Yet it needs to perform repeatedly search due to the limited battery life .
? The time of every sortie equals to its maximum battery life Ti, which con-tains 3 parts:
?aThe time of plane ai reaching searching area from bases Ti
?a The time of plane ai implement search operation Ti
?a The time of plane ai fly from searching area to bases Ti
For convenience, we assume the time of each plane reaching searching zone equals to the time flying back to base and do not consider the time of plane in the base used to replenish fuel. Thus the time of plane ai used to implement
L
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search operation in the whole action Tiduring the whole time T equals to the
?aatotal time for all sorties executive search assignments, namelyTi?LiTi.
6.1.2 Modeling
From the analysis above, it needs to make complete coverage of the searching zone
a?Ti?1Nai?Aixi?STa
Thus, the model can be expressed as follows:
?ST?minT??N?aaT?(1?iL)Aixi??Tii?1?N?s.t?xi?Qa(1?Qa?N)?i?1
6.1.3 Solution of the Model
The model establishes a three-dimensional search for global optimization model with special objective function (Fractional objective function，and the control variable of numerator and denominator is either 0 or 1) and linear constraint, we take them equal to “fractional knapsack problem”. We can calculate the value by using Dinkelbach arithmetic in polynomial time to meet the practical need.
Step 1 The equivalent transformation of the model
To clarify the character of model 3-1 and interpret the solution of it better, we make a equivalent transformation to the objective function:
N?aa?T?p0??(1?iL)AixiAssume that ?Tii?1??q0?ST? （q0?0）
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p0?maxT???q0Then the equivalent transformation is ?N
aa?s.t?xi?Q,1?Q?N??i?1
We can know by the classification of optimizing model mentioned before that model
II belongs to 0-1 programming in the integer programming. Due to its objective func-tion is a nonlinear function and it belongs to nonlinear programming model also with only one special linear equation constraints. Its particularity lies in the control va-riables xi(i?1,?N) and the coefficient ahead is 1, so it is called “gross constraint”.
The following will analyze the value of p0 and control variablexi: according to the expression
?aaTp0??(1?iL)AixiTii?1
N
We know that the value of p0is related to the three positive parameters as follow:
(1)Ai: the search ability of planeai, and Ai?0
aa?a: the time plane needed to fly to and back from the target search sea area, (2)Ti?a?2Da/V?a?0 andTiii(3)Ti: the maximum battery life of planeai, and Ti?0
L?a, that can it In addition, for any plane ai, only when it meets this condition: Ti?TiLLtake part in the search movement. In other words, it should have the ability to fly to and back from the target search sea area within its maximum battery lifeTi. There-fore, we call 3-1 the precondition. For any planeai meeting this precondition, there
L?aa?aaTTis(1?iL)Ai?0. Assume thatai?(1?iL)Ai, thenai?0. Besides, we can always
TiTimeet the following relations by changing the order: